Let \(a,b\)  be positive integers. They are related
to the orders of two orbifold points of \(\mathbb {C}\mathbb {P}^1\) .
The case of \(a = b = 1\)  amounts to the ordinary \(\mathbb {C}\mathbb {P}^1\) .
The equivariant Toda hierarchy of type \((a,b)\) 
can be obtained by imposing the reduction condition
[1]}, [2]}
\(L^a - \nu \log L = \bar{L}^{-b} - \nu \log \bar{L} - \nu \log Q,\) 
